Your question starts out as if the statistical null and alternative hypotheses are what you are interested in, but the penultimate sentence makes me think that you might be more interested in the difference between scientific and statistical hypotheses.

Statistical hypotheses can only be those that are expressible within a statistical model. They typically concern values of parameters within the statistical model. Scientific hypotheses almost invariably concern the real world, and they often do not directly translate into the much more limited universe of the chosen statistical model. Few introductory stats books spend any real time considering what constitutes a statistical model (it can be very complicated) and the trivial examples used have scientific hypotheses so simple that the distinction between model and real-world hypotheses is blurry.

I have written an extensive account of hypothesis and significance testing that includes several sections dealing with the distinction between scientific and statistical hypotheses, as well as the dangers that might come from assuming a match between the statistical model and the real-world scientific concerns: A Reckless Guide to P-values

So, to answer your explicit questions:

• No, statisticians do not always use null and alternative hypotheses. Many statistical methods do not require them.

• It is common practice in some disciplines (and maybe some schools of statistics) to specify the null and alternative hypothesis when a hypothesis test is being used. However, you should note that a hypotheses test requires an explicit alternative for the planning stage (e.g. for sample size determination) but once the data are in hand that alternative is no longer relevant. Many times the post-data alternative can be no more than 'not the null'.

• I'm not sure of the mental heuristic thing, but it does seem possible to me that the beginner courses omit so much detail in the service of simplicity that the word 'hypothesis' loses its already vague meaning.

The rule for the proper formulation of a hypothesis test is that the alternative or research hypothesis is the statement that, if true, is strongly supported by the evidence furnished by the data.

The null hypothesis is generally the complement of the alternative hypothesis. Frequently, it is (or contains) the assumption that you are making about how the data are distributed in order to calculate the test statistic.

Here are a few examples to help you understand how these are properly chosen.

1. Suppose I am an epidemiologist in public health, and I'm investigating whether the incidence of smoking among a certain ethnic group is greater than the population as a whole, and therefore there is a need to target anti-smoking campaigns for this sub-population through greater community outreach and education. From previous studies that have been published in the literature, I find that the incidence among the general population is $p_0$. I can then go about collecting sample data (that's actually the hard part!) to test $$H_0 : p = p_0 \quad \mathrm{vs.} \quad H_a : p > p_0.$$ This is a one-sided binomial proportion test. $H_a$ is the statement that, if it were true, would need to be strongly supported by the data we collected. It is the statement that carries the burden of proof. This is because any conclusion we draw from the test is conditional upon assuming that the null is true: either $H_a$ is accepted, or the test is inconclusive and there is insufficient evidence from the data to suggest $H_a$ is true. The choice of $H_0$ reflects the underlying assumption that there is no difference in the smoking rates of the sub-population compared to the whole.

2. Now suppose I am a researcher investigating a new drug that I believe to be equally effective to an existing standard of treatment, but with fewer side effects and therefore a more desirable safety profile. I would like to demonstrate the equal efficacy by conducting a bioequivalence test. If $\mu_0$ is the mean existing standard treatment effect, then my hypothesis might look like this: $$H_0 : |\mu - \mu_0| \ge \Delta \quad \mathrm{vs.} \quad H_a : |\mu - \mu_0| < \Delta,$$ for some choice of margin $\Delta$ that I consider to be clinically significant. For example, a clinician might say that two treatments are sufficiently bioequivalent if there is less than a $\Delta = 10\%$ difference in treatment effect. Note again that $H_a$ is the statement that carries the burden of proof: the data we collect must strongly support it, in order for us to accept it; otherwise, it could still be true but we don't have the evidence to support the claim.

3. Now suppose I am doing an analysis for a small business owner who sells three products $A$, $B$, $C$. They suspect that there is a statistically significant preference for these three products. Then my hypothesis is $$H_0 : \mu_A = \mu_B = \mu_C \quad \mathrm{vs.} \quad H_a : \exists i \ne j \text{ such that } \mu_i \ne \mu_j.$$ Really, all that $H_a$ is saying is that there are two means that are not equal to each other, which would then suggest that some difference in preference exists.